Strongly minimal edge cover

For a hypergraph [math]H=(V,E) [/math], we call [math]C\subseteq E [/math] an edge cover if [math] \bigcup C =V [/math]. An edge cover [math]C [/math] is strongly minimal if [math]\left|C\setminus C'\right|\leq \left|C'\setminus C\right| [/math] holds for any edge cover [math]C' [/math].

Is it true that if the hypergraph [math]H [/math] has no isolated vertices and all of its hyperedges are finite, then [math]H [/math] admits a strongly minimal edge cover?

Remarks

This conjecture was proposed by R. Aharoni and would imply the Fishbone conjecture by using compactness arguments.